Asymptotic finite-time ruin probability for a bidimensional renewal risk model with constant interest force and dependent subexponential claims

Insurance: Mathematics and Economics 58 (2014) 185–192

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Asymptotic finite-time ruin probability for a bidimensional renewal risk model with constant interest force and dependent subexponential claims Haizhong Yang a , Jinzhu Li b,∗ a

Economic Research Center, Northwestern Polytechnical University, Xi’an 710072, PR China

b

School of Mathematical Science and LPMC, Nankai University, Tianjin 300071, PR China

article

abstract

info

Article history: Received February 2014 Received in revised form June 2014 Accepted 26 July 2014

This paper considers a bidimensional renewal risk model with constant interest force and dependent subexponential claims. Under the assumption that the claim size vectors form a sequence of independent and identically distributed random vectors following a common bivariate Farlie–Gumbel–Morgenstern distribution, we derive for the finite-time ruin probability an explicit asymptotic formula. © 2014 Elsevier B.V. All rights reserved.

MSC: primary 62P05 secondary 62E10 91B30 Keywords: Asymptotics Bidimensional renewal risk model Farlie–Gumbel–Morgenstern distribution Ruin probability Subexponentiality

1. Introduction In this paper, we consider a bidimensional renewal risk model, in which an insurer simultaneously operates two kinds of business sharing a common claim-number process. Concretely speaking, the bidimensional surplus process of the insurer is described as

 N (t )    r (t −τi ) r (t −s) X e i      rt  e C1 (ds)  0−   i=1  U1r (t ) xe    , = + −  rt t     N ( t ) U2r (t ) ye  r (t −s)   r ( t −τ ) e C (ds) i 

t

Yi e

2

0−

t ≥ 0,

i=1

(1.1)

where r ≥ 0 denotes the constant interest force, (x, y) the initial surplus vector, (C1 (t ), C2 (t )) the vector of the total premium accumulated up to time t with the nondecreasing and right continuous components satisfying (C1 (0), C2 (0)) = (0, 0), and

∗

Corresponding author. Tel.: +86 2223501233. E-mail addresses: [email protected] (H. Yang), [email protected] (J. Li).

http://dx.doi.org/10.1016/j.insmatheco.2014.07.007 0167-6687/© 2014 Elsevier B.V. All rights reserved.

{(Xi , Yi ); i ≥ 1} the sequence of claim size vectors whose common arrival times τ1 , τ2 , . . . constitute a renewal claim-number process { N (t ); t ≥ 0} with finite renewal function λ(t ) = EN (t ) = ∞ i=1 P (τi ≤ t ). Throughout this paper, {(Xi , Yi ); i ≥ 1} is assumed to be a sequence of independent and identically distributed (i.i.d.) random vectors with generic vector (X , Y ) whose marginal distribution functions are F = 1 − F on [0, ∞) and G on [0, ∞), respectively. We further assume that {(Xi , Yi ); i ≥ 1} , {(C1 (t ), C2 (t )); t ≥ 0} and {N (t ); t ≥ 0} are mutually independent. Define the finite-time ruin probability of risk model (1.1) as

ψ (x, y; T ) = P ( Tmax ≤ T | (U1r (0), U2r (0)) = (x, y)) ,

T > 0,

where Tmax = inf {t > 0 : max {U1r (t ), U2r (t )} < 0} denotes the ruin time with inf ∅ = ∞ by convention. We aim to seek the precise asymptotic expansion for ψ (x, y; T ) as (x, y) → (∞, ∞) with fixed T .

186

H. Yang, J. Li / Insurance: Mathematics and Economics 58 (2014) 185–192

The asymptotic behavior of finite-time ruin probabilities for risk model (1.1) and its variants (e.g., with r = 0, with constant premium rate, with Poisson claim-number process, or with Brownian perturbation) has been widely investigated in recent years. See Li et al. (2007), Liu et al. (2007), Chen et al. (2011), Zhang and Wang (2012), Chen et al. (2013a,b), Hu and Jiang (2013), and Yin et al. (2013), among others. All of the aforementioned literatures assumed that the claim size vector (X , Y ) consists of the independent components, i.e., {Xi ; i ≥ 1} and {Yi ; i ≥ 1} are independent. However, such a complete independence assumption was proposed mainly for the mathematical tractability rather than the practical relevance. Therefore, to improve this defect of the traditional bidimensional risk models, we assume in the present paper that (X , Y ) follows a bivariate Farlie–Gumbel–Morgenstern (FGM) distribution. Recall that a bivariate FGM distribution with marginal distribution functions F and G is given as

Theorem 2.1. Consider risk model (1.1) in which {(X , Y ), (Xi , Yi ); i ≥ 1} is a sequence of i.i.d. random vectors following a common bivariate FGM distribution (1.2) with θ ∈ (−1, 1]. Let the distribution functions of X and Y be F ∈ S and G ∈ S , respectively. Let T > 0 such that λ(T ) > 0. (i) If θ ∈ (−1, 0] then

  Π (x, y) = F (x)G(y) 1 + θ F (x)G(y) ,

Remark 2.1. In Theorem 2.1, we have to exclude the case of θ = −1, for which our current treatment fails to give a precise asymptotic formula; see, e.g., relations (3.5), (3.10), (3.14), and Remark A.1 below. Actually, in a recent contribution, Chen studied a discrete-time unidimensional risk model with the FGM structure and also excluded such a critical case in the main results due to the similar reason; see Theorems 3.1 and 3.2 of Chen (2011).

θ ∈ [−1, 1].

(1.2)

Trivially, if θ = 0 then (1.2) reduces to a joint distribution function of two independent random variables. Additionally, instead of restricting the claim size distributions to some proper subclasses of the subexponential class or considering the special Poisson claim-number process as done in the existing literatures mentioned above, we greatly weaken the technical assumptions in our framework and conduct the analysis under the general renewal risk model with subexponential claims. In the rest of this paper, Section 2 presents our main result after introducing necessary preliminaries and Section 3 proves the main result after preparing some useful lemmas. Finally, in the Appendix we show a routine (but a bit tedious) proof of Lemma 3.2, which is a crucial result for our purpose. 2. Preliminaries and main result A distribution function V on [0, ∞) is said to belong to the subexponential class, written as V ∈ S , if V (x) > 0 for all x ≥ 0 and the relation lim

x→∞

V n∗ (x) V (x)

=n

holds for all (or, equivalently, for some) n ≥ 2, where V n∗ is the n-fold convolution of V with itself. It is known that if V ∈ S then V ∈ L, which stands for the class of long-tailed distributions characterized by V (x) > 0 for all x ≥ 0 and the relation lim

x→∞

V (x + z ) V (x)

= 1,

z ∈ (−∞, ∞).

One of the most important subclasses of S is the class of distributions with regularly-varying tails. By definition, a distribution function V is said to belong to the class of distributions with regularly-varying tails if V (x) > 0 for all x ≥ 0 and the relation lim

x→∞

V (xz ) V (x)

= z −α ,

z>0

(2.1)

holds for some α ≥ 0. In this case, we write V ∈ R−α . By Theorem 1.5.2 of Bingham et al. (1987), relation (2.1) holds uniformly on all compact z-sets of (0, ∞). Hereafter, all limit relations hold as (x, y) → (∞, ∞) unless otherwise stated. As usual, for two positive bivariate functions f (·, ·) and g (·, ·), we write f . g or g & f if lim sup f /g ≤ 1 and write f ∼ g if both f . g and f & g. To avoid triviality, a nonnegative random variable is always assumed to be nondegenerate at 0. Now, we are ready to present our main result of this paper.

ψ (x, y; T )    r (t +s)   rt      ∼ F xe G ye + F xert G yer (t +s) λ (ds) λ(dt ) s,t ≥0

s+t ≤T

+ (1 + θ )



T

F xert G yert λ(dt ).



 



(2.2)

0−

(ii) If θ ∈ (0, 1] and Eρ N (T ) < ∞ for some ρ > 1 + θ , then relation (2.2) holds.

Remark 2.2. Since the moment generating function of N (T ) is analytic in a neighborhood of 0 (see, e.g., Stein, 1946), there must be some ρ > 1 such that Eρ N (T ) < ∞ and, hence, the conditions of Theorem 2.1(i) and (ii) can be merged as Eρ N (T ) < ∞ for some ρ > 1 + θ + with θ + = max{θ , 0}. Moreover, by the uniformity of relation (2.1) mentioned before, if F ∈ R−α and G ∈ R−α for some α ≥ 0 and {N (t ); t ≥ 0} is a Poisson process with intensity λ > 0 (implying λ(dt ) = λ · dt and Eρ N (T ) < ∞ for all ρ > 0 and T > 0), then the right-hand side of (2.2) can be further expanded to a much more transparent form. Therefore, we have the following corollary immediately. Corollary 2.1. Consider risk model (1.1) in which {(X , Y ), (Xi , Yi ); i ≥ 1} is a sequence of i.i.d. random vectors following a common bivariate FGM distribution (1.2) with θ ∈ (−1, 1]. Let the distribution functions of X and Y be F ∈ R−α and G ∈ R−α for some α ≥ 0, respectively, and let {N (t ); t ≥ 0} be a Poisson process with intensity λ > 0. Then, it holds for any T > 0 that

ψ (x, y; T )   2   λ(1 + θ )  λ   −α rT 2 −2α rT  1 − e + 1 − e F (x)G(y),  2 2 α r 2α r ∼ α, r > 0,     2 2 λ T + λT (1 + θ ) F (x)G(y), α = 0 or r = 0. 3. Proof of Theorem 2.1 3.1. Lemmas The first lemma below is a restatement of Lemma 3.1 of Hao and Tang (2008). Lemma 3.1. Let Z1 , Z2 , . . . be a sequence of independent random variables with distribution functions V1 , V2 , . . . , respectively. Assume that there is a distribution function V ∈ S such that V i (x) ∼ li V (x) with some positive constant li for each i ≥ 1. Then, for each n ≥ 1 and any 0 < a ≤ b < ∞, it holds uniformly for (c1 , . . . , cn ) ∈ [a, b]n that

 P

n  i=1

 ci Zi > x

∼

n  i=1

V i (x/ci ) ,

x → ∞.

H. Yang, J. Li / Insurance: Mathematics and Economics 58 (2014) 185–192

Lemma 3.2 below is a bivariate FGM version of Lemma 3.1. A routine proof of Lemma 3.2 based on the mathematical induction will be given in the Appendix. Lemma 3.2. Let {(X , Y ), (Xi , Yi ); i ≥ 1} be a sequence of nonnegative i.i.d. random vectors following a common bivariate FGM distribution (1.2) with θ ∈ (−1, 1]. Assume that the distribution functions of X and Y are F ∈ S and G ∈ S , respectively. Then, for each n ≥ 1 and any 0 < a ≤ b < ∞, it holds uniformly for (c1 , . . . , cn ) ∈ [a, b]n that

 P

n 

ci Xi > x,

i =1

n 

 cj Yj > y



F (x/ci )G(y/cj ) + (1 + θ )

n  n 

F (x/ci )G(y/ci )

i=1

1≤i̸=j≤n

∼



P(ci Xi > x, cj Yj > y).

(3.1)

I (x, y; c , c ) 11 1 2      ∼(1 + θ ) F (x/c1 ) + F (x/c2 ) G(y/c1 ) + G(y/c2 )      I12 (x, y; c1 , c2 )    ∼ − θ F (x/c ) + 2F (x/c ) G(y/c ) + G(y/c ) 1 2 1 2 I13 (x, y; c1 , c2 )       ∼ − θ F (x/c1 ) + F (x/c2 ) G(y/c1 ) + 2G(y/c2 )      I14 (x, y; c1 , c2 )   ∼θ F (x/c1 ) + 2F (x/c2 ) G(y/c1 ) + 2G(y/c2 ) ,

(3.4)

3 

I1i (x, y; c1 , c2 )

i =1

i =1 j =1

Proof. The second relation in (3.1) is obvious in view of our dependence assumption on {(Xi , Yi ); i ≥ 1}. Note also that the first relation reduces to the second one for n = 1. Next, we only focus on the first relation for n = 2, since the extension to n ≥ 3 follows from the same logic. Motivated from Chen (2011), we decompose the FGM structure as

Π = (1 + θ )FG − θ F 2 G − θ FG2 + θ F 2 G2 . Introduce four independent random variables X1′ , Xˇ 1′ , Y1′ , and Yˇ1′ with distribution functions F , F 2 , G, and G2 , respectively, and let them be independent of (X2 , Y2 ). Then, we have

P (c1 X1 + c2 X2 > x, c1 Y1 + c2 Y2 > y) =:

4 

Ii (x, y; c1 , c2 ),

(3.2)

   I1 (x, y; c1 , c2 ) = (1 + θ )P c1 X1′ + c2 X2 > x, c1 Y1′ + c2 Y2 > y       ′ ′   I2 (x, y; c1 , c2 ) = −θ P c1 Xˇ 1 + c2 X2 > x, c1 Y1 + c2 Y2 > y    I3 (x, y; c1 , c2 ) = −θ P c1 X1′ + c2 X2 > x, c1 Yˇ1′ + c2 Y2 > y        I (x, y; c , c ) = θ P c Xˇ ′ + c X > x, c Yˇ ′ + c Y > y . 4 1 2 1 1 2 2 1 1 2 2 Introduce another four independent random variables X2′ , Xˇ 2′ , Y2′ ,   ′ ′ ′ ′ ′ and Yˇ2 the same as done for X1 , Xˇ 1 , Y1 , Yˇ1 and let them be



X1′ , Xˇ 1′ , Y1′ , Yˇ1′ . Then, we further decompose

I1 (x, y; c1 , c2 ) as I1 (x, y; c1 , c2 ) =: (1 + θ )

4 

I1i (x, y; c1 , c2 ),

+ (1 − 2θ ) F (x/c2 )G(y/c1 ) + (1 − 3θ ) F (x/c2 )G(y/c2 ) =: A(x, y; c1 , c2 ) and

−I14 (x, y; c1 , c2 ) ∼ −θ F (x/c1 )G(y/c1 ) − 2θ F (x/c1 )G(y/c2 ) − 2θ F (x/c2 )G(y/c1 ) − 4θ F (x/c2 )G(y/c2 ) =: B(x, y; c1 , c2 ). Hence, for any 0 < ε < 1, there exist some x0 and y0 such that

(1 − ε)A(x, y; c1 , c2 ) ≤

3 

I1i (x, y; c1 , c2 ) ≤ (1 + ε)A(x, y; c1 , c2 )

and

(1 − ε)B(x, y; c1 , c2 ) ≤ −I14 (x, y; c1 , c2 ) ≤ (1 + ε)B(x, y; c1 , c2 )

where



∼ (1 − θ) F (x/c1 )G(y/c1 ) + (1 − 2θ ) F (x/c1 )G(y/c2 )

i=1

i=1

independent of

relations, we have, uniformly for (c1 , c2 ) ∈ [a, b]2 ,

here and hereafter I12 (x, y; c1 , c2 ), I13 (x, y; c1 , c2 ), and I14 (x, y; c1 , c2 ) are simply understood as 0 when θ = 0. Assume first that θ ∈ (−1, 0]. In this case, by (3.4), it holds uniformly for (c1 , c2 ) ∈ [a, b]2 that

j =1 n

∼

187

hold for x ≥ x0 , y ≥ y0 , and all (c1 , c2 ) ∈ [a, b]2 . It is easy to see that, for any x ≥ 0, y ≥ 0, and (c1 , c2 ) ∈ [a, b]2 , B(x, y; c1 , c2 ) A(x, y; c1 , c2 ) − B(x, y; c1 , c2 )

≤

−4 θ 1+θ

.

(3.5)

Thus, for x ≥ x0 , y ≥ y0 , and all (c1 , c2 ) ∈ [a, b]2 , 4 

I1i (x, y; c1 , c2 )

i =1

≤ (1 + ε) (A(x, y; c1 , c2 ) − B(x, y; c1 , c2 )) + 2ε B(x, y; c1 , c2 )   8θ ≤ 1+ε− ε (A(x, y; c1 , c2 ) − B(x, y; c1 , c2 )) 1+θ and

(3.3)

i=1

where

I11 (x, y; c1 , c2 )  ′   ′   ′   c2 Y2′ > y  = (1 + θ )P c1 X1 + c2 X2 > x P c1 Y1 +     I12 (x, y; c1 , c2 ) = −θ P c1 X ′ + c2 Xˇ ′ > x P c1 Y ′ + c2 Y ′ > y 1 2 1 2   ′   ′  ′ ′   I13 (x, y; c1 , c2 ) = −θ P c1 X1 + c2 X2 > x P c1 Y1 + c2 Yˇ2 > y          I14 (x, y; c1 , c2 ) = θ P c1 X1′ + c2 Xˇ 2′ > x P c1 Y1′ + c2 Yˇ2′ > y .   Note that F ∈ S , G ∈ S and limx→∞ 1 − F 2 (x) /F (x) = limx→∞   1 − G2 (x) /G(x) = 2. Thus, applying Lemma 3.1 to the above

4 

I1i (x, y; c1 , c2 )

i =1

  8θ ≥ 1−ε+ ε (A(x, y; c1 , c2 ) − B(x, y; c1 , c2 )) . 1+θ Noting the arbitrariness of ε gives that, uniformly for (c1 , c2 ) ∈

[ a, b ] 2 , 4 

I1i (x, y; c1 , c2 ) ∼ A(x, y; c1 , c2 ) − B(x, y; c1 , c2 )

i =1

= F (x/c1 )G(y/c1 ) + F (x/c1 )G(y/c2 ) + F (x/c2 )G(y/c1 ) + (1 + θ) F (x/c2 )G(y/c2 ).

(3.6)

188

H. Yang, J. Li / Insurance: Mathematics and Economics 58 (2014) 185–192

Similarly, if θ ∈ (0, 1], applying (3.4) leads to that, uniformly for (c1 , c2 ) ∈ [a, b]2 ,

and I4 (x, y; c1 , c2 ) ∼ θ 4F (x/c1 )G(y/c1 ) + 2F (x/c1 )G(y/c2 )



I11 (x, y; c1 , c2 ) + I14 (x, y; c1 , c2 )

∼ (1 + 2θ ) F (x/c1 )G(y/c1 ) + (1 + 3θ ) F (x/c1 )G(y/c2 ) + (1 + 3θ ) F (x/c2 )G(y/c1 ) + (1 + 5θ ) F (x/c2 )G(y/c2 ) =: A′ (x, y; c1 , c2 )

 + 2F (x/c2 )G(y/c1 ) + (1 + θ) F (x/c2 )G(y/c2 ) . 4

Using the same analysis as in estimating i=1 I1i (x, y; c1 , c2 ) above, i.e., discussing the cases of θ ∈ (−1, 0] and θ ∈ (0, 1], respectively, we have, uniformly for (c1 , c2 ) ∈ [a, b]2 , 4 

and

−I12 (x, y; c1 , c2 ) − I13 (x, y; c1 , c2 ) ∼ 2θ F (x/c1 )G(y/c1 ) + 3θ F (x/c1 )G(y/c2 ) + 3θ F (x/c2 )G(y/c1 ) + 4θ F (x/c2 )G(y/c2 ) =: B′ (x, y; c1 , c2 ). Hence, there exist some x1 and y1 such that

(1 − ε)A′ (x, y; c1 , c2 ) ≤ I11 (x, y; c1 , c2 ) + I14 (x, y; c1 , c2 ) ≤ (1 + ε)A′ (x, y; c1 , c2 ) and

(1 − ε)B′ (x, y; c1 , c2 ) ≤ −I12 (x, y; c1 , c2 ) − I13 (x, y; c1 , c2 ) ≤ (1 + ε)B′ (x, y; c1 , c2 ) hold for x ≥ x1 , y ≥ y1 , and all (c1 , c2 ) ∈ [a, b]2 . It is easy to see that, for any x ≥ 0, y ≥ 0, and (c1 , c2 ) ∈ [a, b]2 , B′ (x, y; c1 , c2 ) A′ (x, y; c1 , c2 ) − B′ (x, y; c1 , c2 )

Ii (x, y; c1 , c2 )

i=1



∼

F (x/ci )G(y/cj ) + (1 + θ )

F (x/ci )G(y/ci ).

i =1

1≤i̸=j≤2

Plugging this relation into (3.2) completes the proof.



For a distribution V ∈ S , the well-known Kesten’s inequality states that, for any ε > 0, there is some constant K = Kε > 0 such that the inequality V n∗ (x) ≤ K (1 + ε)n V (x)

(3.7)

holds for all x ≥ 0 and n ≥ 1; see Athreya and Ney (1972) or Lemma 1.3.5(c) of Embrechts et al. (1997) for its proof. The next lemma gives a counterpart of Kesten’s bound in the bivariate FGM case. Lemma 3.3. Under the conditions of Lemma 3.2, for any ε > 0, there is some constant K > 0 such that the inequality

 P

n 

X i > x,

n 

i=1

≤ 4θ .

2 

 Yj > y

  ≤ K [ 1 + θ + (1 + ε)]n F (x)G(y)

j =1

holds for all x ≥ 0, y ≥ 0, and n ≥ 1.

Thus, for x ≥ x1 , y ≥ y1 , and all (c1 , c2 ) ∈ [a, b]2 , we have

Proof. We first consider the case of θ ∈ (−1, 0]. In this case, for each i ≥ 1, it is clear that Xi and Yi are negatively quadrant dependent, i.e.,

4 

P (Xi > x, Yi > y) ≤ F (x)G(y),

I1i (x, y; c1 , c2 )

i =1

 ≤ (1 + ε) A′ (x, y; c1 , c2 ) − B′ (x, y; c1 , c2 ) + 2ε B′ (x, y; c1 , c2 )   ≤ (1 + ε + 8θ ε) A′ (x, y; c1 , c2 ) − B′ (x, y; c1 , c2 ) 



n 

X i > x,

n 

i=1 4 

n

Hence, by Theorem 1(ii) of Lehmann (1966), i=1 Xi and j=1 Yj are also negatively quadrant dependent for all n ≥ 1, which implies that

P

and

(x, y) ∈ [0, ∞)2 . n

 Yj > y

≤ F n∗ (x)Gn∗ (y)

j =1

for all x ≥ 0, y ≥ 0, and n ≥ 1. Now, we turn to the case of θ ∈ (0, 1]. In this case, it holds that

I1i (x, y; c1 , c2 )

i =1

  ≥ (1 − ε − 8θ ε) A′ (x, y; c1 , c2 ) − B′ (x, y; c1 , c2 ) , which also imply (3.6) by the arbitrariness of ε . Hence, plugging (3.6) into (3.3) yields that, uniformly for (c1 , c2 ) ∈ [a, b]2 ,

P (Xi ∈ dxi , Yi ∈ dyi )

= [1 + θ (1 − 2F (xi )) (1 − 2G (yi ))] F (dxi )G(dyi ) ≤ (1 + θ ) F (dxi )G(dyi ). Hence, for all x ≥ 0, y ≥ 0, and n ≥ 1, we have

I1 (x, y; c1 , c2 )

∼ (1 + θ ) F (x/c1 )G(y/c1 ) + F (x/c1 )G(y/c2 ) + F (x/c2 )G(y/c1 )  + (1 + θ) F (x/c2 )G(y/c2 ) . 

By the same procedures as above, we can derive that, uniformly for (c1 , c2 ) ∈ [a, b]2 , I2 (x, y; c1 , c2 ) ∼ −θ 2F (x/c1 )G(y/c1 ) + 2F (x/c1 )G(y/c2 )

P

 n 

Xi > x,

i=1

 =

 Yj > y

j =1

···

  n

n

x >x in=1 i y >y i=1 i



 + F (x/c2 )G(y/c1 ) + (1 + θ ) F (x/c2 )G(y/c2 ) ,  I3 (x, y; c1 , c2 ) ∼ −θ 2F (x/c1 )G(y/c1 ) + F (x/c1 )G(y/c2 )  + 2F (x/c2 )G(y/c1 ) + (1 + θ ) F (x/c2 )G(y/c2 ) ,

n 

≤ (1 + θ )n

P (Xi ∈ dxi , Yi ∈ dyi )

i=1

 ··· n

  n

x >x in=1 i y >y i=1 i

i=1

= (1 + θ )n F n∗ (x)Gn∗ (y).

  n  F (dxi ) G(dyi ) i =1

H. Yang, J. Li / Insurance: Mathematics and Economics 58 (2014) 185–192

Therefore, whenever θ ∈ (−1, 1], it holds for all x ≥ 0, y ≥ 0, and n ≥ 1 that

 P

n 

X i > x,

i=1

Next, we turn to I2 (x, y; T ). For each n ≥ 1, by conditioning on the values of τ1 , τ2 , . . . , τn+1 and applying Lemma 3.2, we obtain that





n 

Yj > y

≤ 1+θ 

 + n

F n∗ (x)Gn∗ (y).

P

n 

Xi e

−r τ i

n 

> x,

Then, applying (3.7) to

(x) and

Gn∗



(y) completes the proof.



Hereafter, denote by φθ (x, y; T ) the right-hand side of relation (2.2). The next lemma plays a crucial role in the proof of Theorem 2.1.

=

P

N (T ) 

Xi e−r τi > x,

N (T ) 

···

P

Y j e− r τ j > y

n 

−rti

Xi e

> x,

n 

 Yj e

−rtj

>y

j =1

i =1

0≤t1 ≤···≤tn ≤T
=

0≤t1 ≤···≤tn ≤T
× P(τ1 ∈ dt1 , . . . , τn+1 ∈ dtn+1 )    P Xi e−r τi > x, Yj e−r τj > y, N (T ) = n . 1≤i,j≤n

∼ φθ (x, y; T ).

Thus, we have

j=1

i =1

> y, N (T ) = n

× P(τ1 ∈ dt1 , . . . , τn+1 ∈ dtn+1 )      ∼ ··· P Xi e−rti > x, Yj e−rtj > y

Lemma 3.4. Under the conditions of Theorem 2.1, we have



Yj e





1≤i,j≤n



 −r τ j

j =1

i =1

j =1

F n∗

189

I2 (x, y; T )

Proof. Choose some large positive integer N and write

 ∼



N (T )



P

Xi e−r τi > x,

i=1







+

n =N +1

  N n   P

n=1

Xi e−r τi > x,

i =1

n 

 Yj e−r τj > y, N (T ) = n

j =1

(3.8)

≤

P

=

Xi e

−r τ 1

> x,

n 

i=1

∞  

=

T

P

n 

0−

n =N +1

Xi > xert ,

i =1

Yj e

> y, τn ≤ T

n 

=

 Yi > yert

∞  



F xert G yert P (τ1 ∈ dt )

 



j=i+1



P Xi e−r τi > x, Yj e−r τj > y, τi ≤ T



∞  

P Xi e−r τi > x, Yi e−r τi > y, τi ≤ T

+

∞  ∞ 

J1 (x, y; T ) =

T



∞  ∞ 

P Xi e−r τi > x,



P Xi e−r τi > x, Yj e−r τj > y, τj ≤ T





∞  ∞ 

 (3.12)



P Xi e−r (τj +(τi −τj )) > x,

j=1 i=j+1

0−

×



For J1 (x, y; T ), we further write

F xert G yert P (N (T − t ) ≥ n − 1) P (τ1 ∈ dt )

≤K

j =i

j =1



=: J1 (x, y; T ) + J2 (x, y; T ) + J3 (x, y; T ).

T

 

+

n 

i=1 j=i+1

n



+



i=1

0−



 ∞  ∞ i−1  

+

1 + θ + (1 + ε)





j =1 i =j +1

n =N +1

×

P Xi e−r τi > x, Yj e−r τj > y, N (T ) = n

i =1

For any ε > 0, by Lemma 3.3, there exists a constant K such that



(3.11)

∞  ∞  n  

Yj e−r τj > y, N (T ) = n

× P (N (T − t ) ≥ n − 1) P (τ1 ∈ dt ) .

I1 (x, y; T ) ≤ K

1≤i,j≤n

n=N +1

i=1 n=i

 −r τ 1

j =1





I21 (x, y; T )

=

n=N +1



i=1 n=i j=1

I1 (x, y; T ) n 

P Xi e−r τi > x, Yj e−r τj > y, N (T ) = n



Interchanging the order of the sums in I21 (x, y; T ) leads to

For I1 (x, y; T ), we have



−

=: I21 (x, y; T ) − I22 (x, y; T ).

=: I1 (x, y; T ) + I2 (x, y; T ).

∞ 



∞ 

n=1

Y j e− r τ j > y

j=1

∞ 

=

N (T )

∞ 



∞ 



n

1 + θ + (1 + ε)



P (N (T ) ≥ n − 1) .

n=N +1

Recalling Remark 2.2, the conditions of Theorem 2.1 indicate that Eρ N (T ) < ∞ for some ρ > 1 + θ + . Hence, we may choose some ε > 0 sufficiently small such that the series on the right-hand side of (3.9) converges. Therefore, for any 0 < δ < 1, we can find some large positive integer N such that I1 (x, y; T ) ≤ δ



Yj e

(3.9)

=

∞  ∞   j=1 i=j+1



0−

δ φθ (x, y; T ). ≤ 1+θ

 = (3.10)



J1 (x, y; T )

F xert G yert λ(dt )

 

> y, τj + τi − τj ≤ T . 

Since {N (t ); t ≥ 0} is a renewal process, τi − τj is independent of τj and has the same distribution as τi−j . Hence, conditioning on the values of τj and τi − τj yields

T



−r τj

s,t ≥0 s+t ≤T

F xer (t +s) G yert P τi−j ∈ ds P τj ∈ dt



 

 

s,t ≥0 s+t ≤T

F xer (t +s) G yert λ(ds)λ(dt ).



 



 



190

H. Yang, J. Li / Insurance: Mathematics and Economics 58 (2014) 185–192 N (T ) 

Similarly, we have



J 3 ( x, y ; T ) =



F xe

rt

 

G ye

r (t +s)



λ(ds)λ(dt ).

J2 (x, y; T ) ∼ (1 + θ )



F xe

  rt

G ye

 rt

λ(dt ).

(3.13)

Finally, for I22 (x, y; T ), it holds that

P Xi e−r τ1 > x, Yj e−r τ1 > y, τn ≤ T





n=N +1 1≤i,j≤n

=

n=N +1



+



T

P Xi > xert , Yj > yert



n + n P (N (T ) ≥ n − 1) .

n =N +1

Hence, we can find some large positive integer N such that T

F xert G yert λ(dt )



 



0−

≤

δ 1+θ

φθ (x, y; T ).

(3.14)

1−

δ 1+θ



φθ (x, y; T ) . I2 (x, y; T ) . φθ (x, y; T ).

(3.15)

Plugging (3.10) and (3.15) into (3.8) and noting the arbitrariness of δ , we complete the proof of Lemma 3.4. 

 ψ (x, y; T ) = P e−rt Uir (t ) < 0, i = 1, 2, for some 0 < t ≤ T | (U1r (0), U2r (0)) = (x, y)) . On the one hand, it follows from Lemma 3.4 that

ψ (x, y; T ) ≤ P

Xi e

−r τ i

> x,

i=1

N (T ) 

 Yj e

−r τ j

N (T )  i=1

(3.17)

Lemma A.1. For a distribution function V ∈ L, there exists a function l(·) : (0, ∞) −→ (0, ∞) satisfying (i) limx→∞ l(x)/x = 0, (ii) limx→∞ l(x) = ∞, and (iii) l(·) is slowly varying at infinity such that V (c1 x − c2 l(x)) ∼ V (c1 x),

x → ∞,

j=1

(3.16)

x → ∞,

X i e− r τ i −



V (c1 x) ≤ V (c1 x − c2 l(x))

T

e−rs C1 (ds) > x, 0−

(A.1)

holds uniformly on all compact σ -sets of (0, ∞). Since l(·) is slowly varying, it holds uniformly for c1 ∈ [a, b] that l(c1 x) ∼ l(x) as x → ∞. Thus, uniformly for (c1 , c2 ) ∈ [a, b]2 ,

On the other hand,

ψ (x, y; T ) ≥ P

e−rs

The authors are very grateful to the two reviewers for their very thorough reading of the paper and valuable suggestions. The work of Yang was supported by the Scientific Research Program Fund of Shaanxi Provincial Education Department (Grant No. 2013JK0596) and the Statistical Scientific Research Program Funded by National Bureau of Statistics of China (Grant No. 2013LZ42). The work of Li was supported by the National Natural Science Foundation of China (Grant No. 11201245) and the Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20110031120003).

V (x − σ l(x)) ∼ V (x),

>y

∼ φθ (x, y; T ). 

0−

 T × C1 (ds), 0− e−rs C2 (ds) , and in the last step we used Lemma 3.4. Observing the form of φθ (x, y; T ) given in (2.2) and recalling that F ∈ S ⊂ L and G ∈ S ⊂ L, we have, for any u > 0 and v > 0, φθ (x + u, y + v; T )/φθ (x, y; T ) < 1

Proof. In view of Lemma 2.19 and Proposition 2.20(ii) of Foss et al. (2011), there exists a function l∗ (·) : (0, ∞) −→ (0, ∞) satisfying (i) and (ii) such that V (x − l∗ (x)) ∼ V (x) as x → ∞. Furthermore, by Lemma 3.2 of Tang (2008), there exists a slowly varying function l(·) : √ (0, ∞) −→ (0, ∞) satisfying limx→∞ l(x) = ∞ and l(x) ≤ l∗ (x) for all x > 0. It is easy to check that l(·) fulfills (i)–(iii) and

Clearly,

N (T ) 

 T

holds uniformly for (c1 , c2 ) ∈ [a, b]2 for any 0 < a ≤ b < ∞.

3.2. Proof of Theorem 2.1



where H (·, ·) denotes the joint distribution function of

Appendix. A routine proof of Lemma 3.2

Plugging (3.13) and (3.14) into (3.11) yields



φθ (x + u, y + v; T )H (du, dv), 0

Acknowledgments





0

A combination of (3.16) and (3.17) completes the proof of Theorem 2.1. 

0−

I22 (x, y; T ) ≤ δ

∞



ψ (x, y; T ) & φθ (x, y; T ).

× P (N (T − t ) ≥ n − 1) P (τ1 ∈ dt )  T     ≤ F xert G yert P (τ1 ∈ dt ) ×

> y + v H (du, dv)

The above relations and the dominated convergence theorem imply that

0−

n =N +1

2

Yj e

φθ (x + u, y + v; T ) ∼ φθ (x, y; T ).

× P (N (T − t ) ≥ n − 1) P (τ1 ∈ dt )  T ∞      [n(n − 1) + 2n] ≤ F xert G yert

∞  

 −r τ j

and



0−

1≤i=j≤n

1≤i̸=j≤n

∞



I21 (x, y; T ) ∼ φθ (x, y; T ).



Xi e−r τi > x + u,

i=1

∼



N (T ) 

j =1

Plugging J1 (x, y; T ), J2 (x, y; T ), and J3 (x, y; T ) into (3.12) leads to



0

N (T ) 

0−

∞ 

P 0

T





∞



=

For J2 (x, y; T ), since (Xi , Yi ) follows the FGM distribution, it holds that

∞ 

∞



e−rs C2 (ds) > y 0−

j =1

s,t ≥0 s+t ≤T

I22 (x, y; T ) ≤

Yj e−r τj −



T



  l(x) = V c1 x − c2 l (c1 x) l(c1 x)

. V (c1 x − 2c2 l (c1 x)) ∼ V (c1 x),

x → ∞,

H. Yang, J. Li / Insurance: Mathematics and Economics 58 (2014) 185–192

191

where in the last step we used relation (A.1) and its uniformity. This completes the proof of Lemma A.1.  A routine proof of Lemma 3.2. The second relation in (3.1) is obvious in view of our dependence assumption on {(Xi , Yi ); i ≥ 1}. Note also that the first relation reduces to the second one for n = 1. Next, we assume by induction that the first relation in (3.1) holds for some n ≥ 1 and prove it for n + 1. Clearly, it is equivalent to prove that, uniformly for cn := (c1 , . . . , cn ) ∈ [a, b]n ,

 P Xn+1 +

n 

ci Xi > x, Yn+1 +

i=1

∼



n 

 cj Yj > y

j =1

F (x/ci )G(y/cj ) + (1 + θ )

F (x/ci )G(y/ci )

Fig. A.1. Corresponding regions of the nine parts.

i=1

1≤i̸=j≤n

+ F (x)

n 

n 

G(y/cj ) + G(y)

n 

j =1

F (x/ci ) + (1 + θ ) F (x)G(y)

∼ (1 + θ ) F (x)G(y).

i =1

=: Lθ (x, y; cn ) .

(A.2)

We first derive the upper-bound version of (A.2). Let l1 (·) and l2 (·) be the functions specified in Lemma A.1 for F and G, respectively. Then, l(·) = l1 (·) ∧ l2 (·) is such a function for both F and G. According to (Xn+1 , Yn+1 ) in [0, l(x)] × [0, l(y)], [0, l(x)] × (l(y), y − l(y)], [0, l(x)] × (y − l(y), ∞) , (l(x), x − l(x)] × [0, l(y)], (l(x), x − l(x)] × (l(y), y − l(y)], (l(x), x − l(x)] × (y − l(y), ∞), (x − l(x), ∞) × [0, l(y)], (x − l(x), ∞) × (l(y), y − l(y)], and (x − l(x), ∞) × (y − l(y), ∞), we split the probability in relation (A.2) into nine parts and denote them by Ik (x, y; cn ) , 1 ≤ k ≤ 9, respectively; see Fig. A.1 below for the corresponding region of each part. Hereafter, whenever an asymptotic relationship appears it holds uniformly for cn ∈ [a, b]n , and we will not emphasize this any more for the conciseness in writing. For I1 (x, y; cn ), we have

 I1 (x, y; cn ) ≤ P

n 

ci Xi > x − l(x),

i =1

∼



n 

I2 (x, y; cn )

≤P

n 

ci Xi > x − l(x),

i =1

Yn+1 +

n 

 cj Yj > y, l(y) < Yn+1 ≤ y − l(y)

j =1



l(y)

.2



∞

≤

P

n 

ci Xi > x − l(x),

i=1

n  n 

F (x/ci )

i=1 j=1 n

=2

F (x/ci − l(x)/ci )G(y/cj − l(y)/cj )

n 

 cj Yj > (y − v) ∨ l(y) G(dv)

j=1

∞



P cj Yj > (y − v) ∨ l(y) G(dv)



l(y)



n



F (x/ci )P Yn+1 + cj Yj > y,



i=1 j=1

Yn+1 > l(y), cj Yj > l(y) ,



+ (1 + θ)

n 

F (x/ci − l(x)/ci )G(y/ci − l(y)/ci )

F (x/ci )G(y/cj ) + (1 + θ )

n 

F (x/ci )G(y/ci ),

(A.3)

i=1

1≤i̸=j≤n

where in the second and the last steps we used the induction assumption and Lemma A.1, respectively. For I3 (x, y; cn ), we have

 I3 (x, y; cn ) ≤ P

n 

 ci Xi > x − l(x), Yn+1 > y − l(y)

i =1

 =P

n 





    ≤ P Yn+1 + cj Yj > y − P Yn+1 > y, cj Yj ≤ l(y)   − P cj Yj > y, Yn+1 ≤ l(y)   = o(1) G(y) + G(y/cj ) , y → ∞. I2 (x, y; cn ) = o(1)Lθ (x, y; cn ) .

ci Xi > x − l(x) G(y − l(y))

i =1

∼ G(y)

P Yn+1 + cj Yj > y, Yn+1 > l(y), cj Yj > l(y)

F (x/ci ),

(A.4)

where in the last step we used Lemmas 3.1 and A.1. Symmetrically,

I4 (x, y; cn ) = o(1)Lθ (x, y; cn ) .

G(y/cj ).

(A.5)

(A.10)

For I6 (x, y; cn ), we have

 I7 (x, y; cn ) ∼ F (x)

(A.9)

Symmetrically,

i =1

n 

(A.8)

Plugging the above estimations into (A.7), we obtain that



n 

(A.7)

where in the third step we used the induction assumption and Lemma A.1. For each 1 ≤ j ≤ n, it follows from Lemma 3.1 that

i=1





cj Yj > y − l(y)

j=1

(A.6)

Now, we turn to the other five parts, which will be proved to be asymptotically negligible compared with Lθ (x, y; cn ). For I2 (x, y; cn ), we have



1≤i̸=j≤n

∼

I9 (x, y; cn ) ≤ P (Xn+1 > x − l(x), Yn+1 > y − l(y))

I6 (x, y; cn ) ≤ P Xn+1 +

n 

ci Xi > x,

i=1

j =1

For I9 (x, y; cn ), noting that (Xn+1 , Yn+1 ) follows the FGM distribution and using Lemma A.1, we have

 l(x) < Xn+1 ≤ x − l(x), Yn+1 > y − l(y)

192

H. Yang, J. Li / Insurance: Mathematics and Economics 58 (2014) 185–192

≤ 2G(y − l(y)) ∼ 2G(y)

n  

= 2G(y)

n 

P

l(x)

n 





ci Xi > (x − u) ∨ l(x) F (du)

P Xn+1 +

ci Xi > x, Yn+1 +

i =1

n 

 cj Yj > y

j =1

≥ P ((E1 ∪ E2 ) ∩ (E3 ∪ E4 ))

P (ci Xi > (x − u) ∨ l(x)) F (du)

≥ P (E1 E3 ) + P (E1 ) P (E4 ) + P (E2 ) P (E3 ) + P (E2 E4 ) − P (E2 ) P (E1 E3 ) − P (E1 ) P (E2 E4 )

P (Xn+1 + ci Xi > x, Xn+1 > l(x), ci Xi > l(x))

− P (E1 E3 ) P (E4 ) − P (E3 ) P (E2 E4 )

i =1

 = o(1) G(y)F (x) + G(y)

n 

=:

 F (x/ci ) (A.11)

where the second step follows from the obvious relation Π (du, dv) ≤ 2F (du)G(dv), the third step from Lemmas 3.1 and A.1, and the fifth step from the counterpart of relation (A.8). Symmetrically, I8 (x, y; cn ) = o(1)Lθ (x, y; cn ) .

(A.12)

Finally, by the similar idea as in dealing with I6 (x, y; cn ) and I8 (x, y; cn ), we have I5 (x, y; cn ) ≤ 2

∞ l(y)



∞



l(x)

P

Jk (x, y; cn ) −

8 

Jk (x, y; cn ) .

k=5

By Lemma 3.1 and the induction assumption, it is easy to see that

= o(1)Lθ (x, y; cn ) ,



4  k=1

i =1

n 

n 

i =1

∞ l(x)

i =1



∞



n 

ci Xi > (x − u) ∨ l(x),

4 

Jk (x, y; cn ) ∼ Lθ (x, y; cn )

k=1

and 8 

Jk (x, y; cn ) = o(1)Lθ (x, y; cn ) .

(A.14)

k=5

These estimations imply the lower-bound version of (A.2) and, hence, complete the proof of Lemma 3.2.  Remark A.1. The validities of relations (A.9)–(A.14) essentially depend on θ ̸= −1, which guarantees that no term in Lθ (x, y; cn ) is vanished.

i =1

 cj Yj > (y − v) ∨ l(y) F (du)G(dv)

j =1

.4

n  n   i=1 j=1

∞ l(y)

References

∞



l(x)

P (ci Xi > (x − u) ∨ l(x))

 × P cj Yj > (y − v) ∨ l(y) F (du)G(dv) 

=4

n  n 

P (Xn+1 + ci Xi > x, Xn+1 > l(x), ci Xi > l(x))

i=1 j=1

  × P Yn+1 + cj Yj > y, Yn+1 > l(y), cj Yj > l(y) = o(1)

n  n  

F (x) + F (x/ci )

G(y) + G(y/cj )





i=1 j=1

= o(1)Lθ (x, y; cn ) .

(A.13)

A combination of (A.3)–(A.6) and (A.9)–(A.13) gives the upperbound version of (A.2). The corresponding lower-bound version of (A.2) can be easily established. Define four events as follows: E1 = (Xn+1 > x) ,

E2 =

 n 

 ci Xi > x ,

i=1

 E3 = (Yn+1 > y) ,

E4 =

n 

 cj Yj > y .

j =1

Then, noting our dependence assumption on {(Xi , Yi ); i ≥ 1}, we have

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